1 - 5 of 5
Number of results to display per page
Search Results
2. Estimates for the commutator of bilinear Fourier multiplier
- Creator:
- Hu, Guoen and Yi, Wentan
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- bilinear Fourier multiplier operator, commutator, and Hardy space
- Language:
- English
- Description:
- Let $b_1, b_2 \in {\rm BMO}(\mathbb {R}^n)$ and $T_{\sigma }$ be a bilinear Fourier multiplier operator with associated multiplier $\sigma $ satisfying the Sobolev regularity that $\sup _{\kappa \in \mathbb {Z}} \|\sigma _{\kappa }\| _{W^{s_1,s_2}(\mathbb {R}^{2n})}<\infty $ for some $s_1,s_2\in (n/2,n]$. In this paper, the behavior on $L^{p_1}(\mathbb {R}^n)\times L^{p_2}(\mathbb {R}^n)$ $(p_1,p_2\in (1,\infty ))$, on $H^1(\mathbb {R}^n)\times L^{p_2}(\mathbb {R}^n)$ $(p_2\in [2,\infty ))$, and on $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$, is considered for the commutator $T_{{\sigma }, \vec {b}} $ defined by $$ \begin {aligned} T_{\sigma ,\vec {b}} (f_1,f_2) (x)=&b_1(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(b_1f_1, f_2)(x) &+ b_2(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(f_1, b_2f_2)(x) . \end {aligned} $$ By kernel estimates of the bilinear Fourier multiplier operators and employing some techniques in the theory of bilinear singular integral operators, it is proved that these mapping properties are very similar to those of the bilinear Fourier multiplier operator which were established by Miyachi and Tomita.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. On the $H^{p}$-$L^{q}$ boundedness of some fractional integral operators
- Creator:
- Rocha, Pablo and Urciuolo, Marta
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- integral operator and Hardy space
- Language:
- English
- Description:
- Let $A_{1},\dots ,A_{m}$ be $n\times n$ real matrices such that for each $1\leq i\leq m,$ $A_{i}$ is invertible and $A_{i}-A_{j}$ is invertible for $i\neq j$. In this paper we study integral operators of the form $$ Tf( x) =\int k_{1}( x-A_{1}y) k_{2}( x-A_{2}y) \dots k_{m}( x-A_{m}y) f( y) {\rm d} y, $$ $k_{i}( y) =\sum _{j\in \mathbb Z}2^{jn/{q_{i}}}\varphi _{i,j}( 2^{j}y) $, $1\leq q_{i}<\infty,$ $1/{q_{1}}+1/{q_{2}}+\dots +1/{q_{m}}=1-r,$ $0\leq r<1,$ and $\varphi _{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T\colon H^{p}( \mathbb {R} ^{n}) \rightarrow L^{q} (\mathbb {R}^{n}) $ for $ 0<p<1/{r}$ and $1/{q}=1/{p}-r.$ We also show that we can not expect the $H^{p}$-$H^{q}$ boundedness of this kind of operators.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4. On the reflexivity of subspaces of Toeplitz operators on the Hardy space on the upper half-plane
- Creator:
- Młocek, Wojciech and Ptak, Marek
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- reflexive subspace, transitive subspace, Toeplitz operator, Hardy space, and upper half-plane
- Language:
- English
- Description:
- The reflexivity and transitivity of subspaces of Toeplitz operators on the Hardy space on the upper half-plane are investigated. The dichotomic behavior (transitive or reflexive) of these subspaces is shown. It refers to the similar dichotomic behavior for subspaces of Toeplitz operators on the Hardy space on the unit disc. The isomorphism between the Hardy spaces on the unit disc and the upper half-plane is used. To keep weak* homeomorphism between $L^\infty $ spaces on the unit circle and the real line we redefine the classical isomorphism between $L^1$ spaces.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
5. Some estimates for commutators of Riesz transform associated with Schrödinger type operators
- Creator:
- Liu, Yu, Zhang, Jing, Sheng, Jie-Lai, and Wang, Li-Juan
- Format:
- print, bez média, and svazek
- Type:
- model:article and TEXT
- Subject:
- mathematics, Schrödinger equation, commutator, Hardy space, everse Hölder inequality, Riesz transform, Schrödinger operator, Schrödinger type operator, 13, and 51
- Language:
- English
- Description:
- Let L1 = −Δ + V be a Schrödinger operator and let L2 = (−Δ)2 + V2 be a Schrödinger type operator on \mathbb{R}^{n}\left ( n\geqslant 5 \right ) where V≠ 0 is a nonnegative potential belonging to certain reverse Hölder class Bs for s\geqslant n/2. The Hardy type space H_{L2}^{1} is defined in terms of the maximal function with respect to the semigroup \left\{ {{e^{ - t{L_2}}}} \right\} and it is identical to the Hardy space H_{L2}^{1} established by Dziubański and Zienkiewicz. In this article, we prove the Lp-boundedness of the commutator Rb = bRf - R(bf) generated by the Riesz transform R = {\nabla ^2}L_2^{ - 1/2} , where b \in BM{O_\theta }(\varrho ) , which is larger than the space BMO\left (\mathbb{R}^{n} \right ). Moreover, we prove that Rb is bounded from the Hardy space H_{L2}^{1} into weak L_{weak}^1 (\mathbb{R}^n )., Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang., and Obsahuje seznam literatury
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public